Ongoing Projects
Project-1
Title: k-plane Transforms and their properties on Manifolds
Description: We will be looking at the X-Ray transform, its generalization, the Radon transform, and its further generalization, the k-plane transforms. A lot of research has been done in the field of reconstructing a function using the k-plane transforms on the Euclidean spaces. We will be interested in some properties of these transforms and their behaviour on (smooth) manifolds and spaces of constant curvature.
This project is a part of the PhD thesis, and collaboration is not possible for now.
Project-2
Title: Rigidity of Angles in Determining the Geometry
Description: We are interested in knowing if angles defined by an inner product on a vector space completely determines the inner product. Particularly, we ask if two inner products on a vector space define the same angles, then are the two inner products same. We will also be interested to ask similar questions for Riemannian manifolds (where a Riemannian metric is a choice of smoothly varying inner products on tangent spaces).
Project-3
Title: Topology of Non-Triangular Metric Spaces
Description: Recently, in 2020, the concept of non-triangular metric spaces was given. This new concept generalizes metric spaces by omitting the triangle inequality axiom. We will be interested in studying the topology generated by this new definition of "distance function". Eventually, we could answer questions about compactness, connectedness, continuity of functions and even the metrizability of non-triangular spaces.
The paper on "Topology of Non-Triangular Metric Spaces and Related Fixed Point Results" has been published in Filomat.
This project is currently on hold due to PhD committments.
Project-4
Title: Fixed Point Theory
Description: We will be interested in studying the existence and uniqueness of Fixed Points of various self-maps on metric and normed linear spaces. As an extension, we can also study maps on spaces more general than metric spaces or iterative procedures of approximation of fixed points.